Node-Capacitated Ring Routing

We consider the node-capacitated routing problem in an undirected ring network along with its fractional relaxation, the node-capacitated multicommodity flow problem. For the feasibility problem, Farkas’ lemma provides a characterization for general undirected graphs asserting roughly that there exists such a flow if and only if the so-called distance inequality holds for every choice of distance functions arising from non-negative node-weigths. For rings this (straightforward) result will be improved in two ways. We prove that, independent of the integrality of node-capacities, it suffices to require the distance inequality only for distances arising from (0-1-2)-valued node-weights, a requirement which will be called the double-cut condition. Moreover, for integer-valued node-capacities, the double-cut condition implies the existence of a half-integral multicommodity flow. In this case there is even an integervalued multicommodity flow which violates each node-capacity by at most one. Our approach gives rise to a combinatorial, strongly polynomial algorithm to compute either a violating double-cut or a node-capacitated multicommodity flow. A relation of the problem to its edge-capacitated counterpart will also be explained.